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Stability of mean curvature flow solitons in warped product spaces

Abstract

In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. We also prove some non-existence results for solitons as applications of a comparison principle which suits well the structure of the diffusion elliptic operator associated to the weighted measures we are considering.

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Acknowledgements

The authors would like to thank the anonymous referees for reading the manuscript in great detail and for giving several valuable suggestions and useful comments which improved the paper.

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Correspondence to Luis J. Alías.

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This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia.

Luis J. Alías and Marco Rigoli were partially supported by MICINN/FEDER project PGC2018-097046-B-I00 and Fundación Séneca project 19901/GERM/15, Spain.

Jorge H.S. de Lira was partially supported by PROEX/CAPES and PQ-CNPq \(\#\) 307410/2018-8.

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Alías, L.J., de Lira, J.H.S. & Rigoli, M. Stability of mean curvature flow solitons in warped product spaces. Rev Mat Complut 35, 287–309 (2022). https://doi.org/10.1007/s13163-021-00394-y

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  • DOI: https://doi.org/10.1007/s13163-021-00394-y

Keywords

  • Mean curvature flow soliton
  • Stability
  • Spectral theory of elliptic operators

Mathematics Subject Classification

  • 53C42
  • 53C21
  • 49Q20